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User talk:Hyp cos/Dropping hydra
As I understand the limit ordinal for DAN, R-function, and Dropping hydra are the same? According to my approximate analysis Cacth-function if it were defined - this ordinal we could write C(Ωω). Perhaps after this there is a complete convergence FHG and SGH? It seems that this ordinal is PTO for some theory. I propose to call it Hypcos ordinal.Scorcher007 (talk) 05:55, January 4, 2018 (UTC) :Limits of DAN, R function and dropping hydra are the same. :Catching function is sensetive to fundamental sequences, and it can't be defined without refering FS. I don't know what's next catching point after the dropping hydra limit, because I have got stuck here for about 4 years. :The theory may be \(\Pi^1_2\text{ - CA}_0\), \(\Pi^1_2\text{ - TR}_0\) or second-order arithmetic, I guess. {hyp/^,cos} (talk) 09:38, January 4, 2018 (UTC) :1) If Buchholz hydra ordinal is PTO of \(\Pi^1_1\text{ - CA}_0\). Possibly Dropping hydra ordinal is PTO of \(\Pi^1_2\text{ - CA}_0\). Besides 1st solution of Catching function C(0) is OCF(Ωω) or \(\Pi^1_1\text{ - CA}_0\). And C(Ωω) equal Dropping ordinal. :By the way, as for the Catching function, you can define a Catching function without using ψ or other OCF: :C(n) = n-catching point of FGH and SGH C(Ω) = α↦C(α) C(Ω+n) = n-catching point after α↦C(α) C(Ω×2) = n-fixed point of α↦C(α) C1(n) = FS of C(n) on Ω C1(Ω) = α↦C1(C(α)) C1(Ω2) = α↦C1(α) C2(n) = FS of C(n) on Ω2 :and so on up to C(Ωω) :Where''' FS of C(n) on Ω''' means "fundamental sequences of ordinal C(n) in some well-ordering ordinal notation capable of describing it, applied to Ω". :For example, the same not clear definition has Rayo's function, but it is still considered a definite. :2) Taranovsky wrote somewhere that C(C(C(Ω32,0),0),0) in his notation equal \(\Pi^1_2\text{ - TR}_0\). It means if Dropping ordinal is PTO of \(\Pi^1_2\text{ - CA}_0\), then Dropping ordinal < C(C(C(Ω32,0),0),0) ? :May be C(ΩΩΩΩ...) is \(\Pi^1_2\text{ - TR}_0\)? Scorcher007 (talk) 10:06, January 13, 2018 (UTC) :1) The limit of Buchholz hydra is \(\psi(\varepsilon_{\Omega_\omega+1})\), PTO of \(\Pi^1_1\text{ - CA}_0+\text{BI}\), while the limit of Buchholz hydra without the \(\omega\) label is \(\Pi^1_1\text{ - CA}_0\). :"fundamental sequences of ordinal C(n) in some well-ordering ordinal notation capable of describing it, applied to Ω" still refers some other ordinal notations. :2) I have compared Taranovsky's ordinal notation (TON) with pDAN and sDAN. TON seems to be not so strong as I expected when it goes beyond C(C(Ω22+C(Ω2+C(Ω2,C(Ω2,C(Ω22,0))),0),0),0). I have some intuitive feeling that TON might be as strong as DAN. {hyp/^,cos} (talk) 15:29, January 13, 2018 (UTC) :Then it is obtained similar to Buchholz hydra with ω: :according to my assumption: :C(0) = {1,,1,2} = {1{1,,,2}1,2} :C(C1(Ω)) = {1{1{1,,,1,2}2,,,2}2} = {1,,,1,2} = {1{1,,,,2}1,2} :C(C1(C2(Ω2))) = {1{1{1,,,,1,2}2,,,,2}2} = {1,,,,1,2} = {1{1,,,,,2}1,2} :C(C1(C2(C3(Ω3)))) = {1{1{1,,,,,1,2}2,,,,,2}2} = {1,,,,,1,2} = {1{1,,,,,,2}1,2} :C(C1(C2(C3(C4(Ω4))))) = {1{1{1,,,,,,1,2}2,,,,,,2}2} = {1,,,,,,1,2} = {1{1,,,,,,,2}1,2} :C(Ωω) = C(C1(C2(C3(C4(C5(...)))))) = {1,,, ... ,,,1,2} with n+2 "," or {1{1,,, ... ,,,2}1,2} with n+3 "," :is \(\Pi^1_2\text{ - CA}_0\) :but we can describe {1{1{1,,, ... ,,,1,2}2,,, ... ,,,2}2} with n+3 "," or {1,,, ... ,,,1,2} with n+3 :like C(Cω(Ωω)) :is \(\Pi^1_2\text{ - CA}_0+\text{BI}\) :And it's Dropping hydra ordinal with ω and limit of DAN. :I guess if NDAN worked, then it has limit ordinal = C(ΩΩΩΩ...) = \(\Pi^1_2\text{ - TR}_0\) :It is interesting that in his old works Taranovsky assumed his comparisons according to a similar scheme: :C(Ω+1,0) = εω :C(Ω,C(Ω+1,0)) = εω+1 :C(Ω×2,0) = εεεε... :C(C(Ω2+1,0),0) = ψ(Ωω) = П11-CA0 :C(C(Ω2,C(Ω2+1,0)),0) = ψ(εΩω+1) = П11-CA-BI :C(C(Ω2×2,0),0) = ψ(ΩΩΩΩ...) = П11-TR0 :C(C(C(Ω3+1,0),0),0) = П12-CA0 :C(C(C(Ω3,C(Ω3+1,0)),0),0) = П12-CA-BI :C(C(C(Ω3×2,0),0),0) = П12-TR0 :C(C(C(C(Ω4+1,0),0),0) = П13-CA0 :C(C(C(C(Ω4,C(Ω4+1,0)),0),0),0) = П13-CA-BI :C(C(C(C(Ω4×2,0),0),0) = П13-TR0 :C(...C(C(C(C(Ωω+1,0),0),0)...,0) = Z2 :Scorcher007 (talk) 16:41, January 13, 2018 (UTC) :Actually, kind of notation does not matter for power of fundamental sequences. We can apply TON or you Dropping hydra's ordinal notation, power of fundamental sequences for Catching function will be the same. I slightly corrected my intuitive analysis for Catching function. Now it looks like this http://lihachevss.ru/catchfunction.html :Scorcher007 (talk) 06:06, January 14, 2018 (UTC) :Let's focus on the problem of catching function. I mean, it can't be defined without refering other ordinal notations. Let \(\alpha\) be such ordinal that \(C(\alpha)\) reaches the supremum of the limits of normal OCF's, dropping hydra, higher strong array notation (if they were defined), TON, and so on. Then how to define \(C(\alpha+1)\), \(C(\alpha+\Omega)\), \(C(\alpha+\Omega_\omega)\), \(C(C_\alpha(\alpha))\), and such things? {hyp/^,cos} (talk) 10:23, January 14, 2018 (UTC) :Firstly, as I understand, C-function is not ordinal notation. I agree that the C-function is meaningful only on those scales, as long as there is a ordinal notation that can give it a fundamental sequence. But can we define all ordinal notations as a mathematical object? Something like "Any ordinal notations that can be defined in Z2". And use like reflection of V or L in set theory. Then the definition of a C-function can be made universal. :Scorcher007 (talk) 11:13, January 14, 2018 (UTC)